Problem 19 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=3 \sin ^{2} x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = 6x\sin(2x^2) \).

Step 1: Identify the Function Type

The given function is \( f(x) = 3 \sin^2(x^2) \). Recognize that this is a composition of functions: it's a constant multiple of a squared sine function of \( x^2 \).

Step 2: Differentiate the Outer Function

Apply the chain rule to differentiate the outer function \( 3 \sin^2(u) \), where \( u = x^2 \). The derivative of \( \ \sin^2(u) \) with respect to \( u \) is \( 2 \sin(u) \cos(u) \). Thus, the derivative of \( 3\sin^2(u) \) with respect to \( u \) is \( 3 \times 2 \sin(u) \cos(u) = 6 \sin(u) \cos(u) \).

Step 3: Differentiate the Inner Function

The inner function is \( u = x^2 \). The derivative of \( x^2 \) with respect to \( x \) is \( 2x \).

Step 4: Apply the Chain Rule

To find the derivative of \( f(x) = 3 \sin^2(x^2) \), use the chain rule. Multiply the derivative of the outer part by the derivative of the inner part: $$ f'(x) = 6 \sin(x^2) \cos(x^2) \cdot 2x = 12x \sin(x^2) \cos(x^2). $$

Step 5: Simplify the Expression

Notice that \( 2 \sin(x) \cos(x) \) is equivalent to \( \sin(2x) \) by the double-angle identity for sine. Therefore, \( 12x \sin(x^2) \cos(x^2) \) simplifies to \( 6x \sin(2x^2) \). Thus, the derivative is: $$ f'(x) = 6x\sin(2x^2). $$

Key Concepts

Chain Rule

The chain rule is an essential technique in calculus, especially for finding the derivative of composite functions. It helps us differentiate a function composed of two or more functions. Conceptually, the chain rule can be expressed as:

• If you have a function that can be seen as a composition of two functions, say \( h(x) = f(g(x)) \), then the derivative \( h'(x) \) can be found by multiplying the derivative of the outer function \( f \) evaluated at \( g(x) \), by the derivative of the inner function \( g(x) \).

This is written as: \[ h'(x) = f'(g(x)) imes g'(x) \] Using the chain rule is like peeling an onion, you differentiate layer by layer. Each application of the chain rule uncovers another layer. In the original problem, we applied the chain rule when we differentiated \( \sin^2(x^2) \) by recognizing \( \ u = x^2 \) and first differentiating the outer function \( \sin^2(u) \), followed by the inner \( x^2 \).

Trigonometric Differentiation

Trigonometric differentiation refers to the process of finding derivatives of trigonometric functions. The most fundamental trigonometric functions include \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \), and they have specific derivatives:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

In our exercise, trigonometric differentiation was used when finding the derivative of \( \sin^2(u) \). Recognizing \( \sin^2(u) \) uses a trigonometric identity, where we originally found its derivative using \( 2 \sin(u) \cos(u) \). Additionally, the double-angle identity \( 2 \sin(x) \cos(x) = \sin(2x) \) simplified the final result of the derivative.

Derivative

A derivative represents the rate of change or slope of a function at any given point. When we take the derivative of a function \( f(x) \), we are essentially asking how \( f(x) \) changes as \( x \) changes. This is notated as \( f'(x) \) or \( \frac{df}{dx} \).

• For example, if \( f(x) = x^2 \), then the derivative, \( f'(x) = 2x \), tells us that the slope of \( f(x) \) at any point \( x \) is \( 2x \).

In this particular problem, we sought the derivative of the function \( f(x) = 3 \sin^2(x^2) \). The process involved differentiating each part of the function individually before combining them according to the rules for derivatives, such as the chain rule. Calculating derivatives is crucial in understanding how dynamic systems behave and change.

Composition of Functions

Composition of functions is a fundamental concept where one function is applied to the results of another function. If you have two functions \( f \) and \( g \), the composition is written as \( f(g(x)) \), meaning \( g(x) \) is evaluated first, and then the resulting value is used as the input for \( f \).

• In the exercise, the function \( f(x) = 3 \sin^2(x^2) \) is a composite function because it combines both \( \sin(x) \) and the square of another function \( x^2 \). Here, \( \sin^2(x^2) \) illustrates both composition and the need for careful differentiation.

Understanding the composition of functions is helpful for effectively applying the chain rule. It involves recognizing which function is "inside" and which is "outside" so that we can tackle them in a step-by-step manner. Through composition and chain rule application, we can simplify complex derivative equations to more manageable forms, giving us clearer insights into the behavior of functions.